Answer the following questions, fully justifying your answers:

a) Consider the probability that the stock market index will increase in value next year. The rational expectation for the value of this probability is 45% and the corresponding 95% confidence interval is (30%,65%). Additionally, consider an investor who doesn't know probability and suffers from the over-precision bias. This investor is asked to provide an estimate for probability as well as a 95% confidence interval for it. What are plausible values for these?

b) Consider an individual who will be retiring in two years, i.e., at time t + 2. His savings are invested in a mutual fund. Every year, the value of his portfolio increases by G with probability p and otherwise decreases by L, where G = L. The individual's preferences can be represented by Prospect Theory. Specifically, his value function for gains and losses relative to the reference point (the value of the savings at time t) is

() = for 0

for < 0

where > 1. Furthermore, his probability weighting function is () = , and his time discount factor is = 0.5. The individual wants to maximize his utility from wealth at retirement. If, at the end of the first year (time + 1), the individual observes that his investments have gone down by , does he optimally choose to stay invested in the fund or not? Discuss. Also discuss how your answer would change if

() = x^a for 0

()^a for < 0

for a > 0.